Hierarchical model of the magnetorheological response in isotropic and anisotropic magnetic elastomers.

We present a hierarchical theoretical model and experimental validation of magnetorheological hysteresis in soft magnetoactive elastomers filled with magnetic iron microparticles. The study encompasses both isotropic composites (polymerized without a magnetic field) and anisotropic composites (cured under a magnetic field, yielding "frozen" chain-like microstructures). A key theoretical advance is the quantitative description of field-cycle hysteresis: during field increase, magnetic agglomerates assemble into chains via a hierarchical aggregation process; during field decrease, chain disintegration follows a distinct pathway governed by elastic rupture forces, producing a hysteresis loop in the shear modulus versus magnetic field dependence. For anisotropic elastomers, the model accounts for pre-formed chains established during curing and predicts their limited reconfiguration under subsequent field application. Experimentally, an external field up to 0.8 T induces a >10-fold increase in shear storage modulus, with anisotropic samples exhibiting a substantially stronger response. The model, based on primary agglomerates forming field-aligned chains, provides quantitative agreement with measured moduli across magnetic field cycles for both composite types. This work establishes the first self-consistent theory capturing both anisotropic composite response and magnetorheological hysteresis within a unified hierarchical framework.